Third order Approximation on Icosahedral Great Circle Grids on the Sphere. J. Steppeler, P. Ripodas DWD Langen 2006

Transcription

1 Tird order Approximation on Icosaedral Great Circle Grids on te Spere J. Steppeler, P. Ripodas DWD Langen 2006

2 Deasirable features of discretisation metods on te spere Great circle divisions of te spere: cube, 4-body, isocaedron Icosaedral stencils Overview on discretisation options Great circle grids and ig order approximation How to make a grid Boundaries and running a model Semi implicit integration on te Isocaedron and domain decomposition Plan of Lecture Some Result of a RK4 / spatially tird order sallow water model

3 Desirable features of discretisations on te spere Problems appearing in te same way on te isocaedron and for limited area models Observation of approximation conditions: smootness (for tird order scemes), Smoot pysics interface, smoot orograpy (d<dz) or z-coordinate) Positivity of advection: flux correction Nesting option: Skamarock metod

4 Desirable features of discretisations on te spere Special callenges for icosaedral grids Nonydrostatic Accuracy: Order 3 or iger in space [and time] Conservation: mass, energy Efficiency (computer time and development time) Ability to incorporate developments for n models

5 Te Baumgardner principle (Rules of good beaviour on triangles Proven for o2, not yet for o3) No global coordinate Keep approximation order at grid interfaces Te resulting sceme as no problems carrying plane discretisations to te spere

6 Creation of basic polygon Divide te angle of 360 degrees at bot poles into NP parts. Create equally sided romboids for eac of te 2*NP divisions. Te computational grid is obtained by subdividing te romboids. Many metods to create te grid lead to irregular lines of points. Here te subdivision by great circles is proposed. Te resulting grid is called great circle grid.

7 Romboidal divisions of te spere NP=3 NP=4 NP=5 Cube 4-body Isocaedron

8 Metods to create grid stencils Te finite volume metod is based on computing fluxes troug te sides of te dual grid cell. Wen assuming amplitudes on te edges of te triangles te dual cell is te exagon. Baumgardner metod: define te grid stencil and compute te polynomial approximating te amplitudes at te stencil in te least square sense and differentiate in order to get an approximation for te derivative. A different local polar coordinate system is used at eac point. For a great circle grid take normal finite difference expressions along a line of grid points. Tis metod is suitable to generalise a limited area model defined on a structured grid

9 Edges grid Grid Stencils Baumgardner Edges grid Order3 Order 2 Redundancy 19:9 Redundancy 6:5 or 5:5

10 Grid Stencils Baumgardner Area grid Area grid Order3 Order 1 (Finite Volume) Redundancy 13:9 Redundancy 4:3

11 Summary Baumgardner grid stencils Order 2 on edges grid is used Order 3 on edges grid as large grid redundancy Te area grid allows order 1 (2 wit superconvergence) and order 3 Te order 3 approximation on te area grid as an irregular resolution (1:2) for te plane wave.

12 Discretisation Options, Based on Interpolation Finite Volumes: Bonaventura coice, best on regularised grids, conservation possible, often low order, tested by Ringler and Steppeler Baumgardner: suitable for somewat irregular grids, tested for order 2 Baumgardner Order2: Amplitudes on edges; small grid redundancy Baumgardner Order3: Amplitudes on triangle surfaces, very irregula grid for plane waves, yet untested, (some grid redundancy) Great circle grids: very similar to limited area discretisations, order 2,3 easily possible, RK, SI, SL, adaptation of all local developments easy (grid redundancy no problem) Tiled grids: very uniform grids (~1%), less elegant look, spectral elements possible)

13 Grid matcing at most boundaries nearly ortogonal Rooftile Grids Interpolation O3 for boundary values for 100 points per tile grid irregularity is 1 %

14 Rooftile Grids, 4-Body Triangles are used to matc areas (implying irregular saped cells or double grid covering). All oter boundaries matc

15 Structured (index i,j) Quasi regular grids Eac line of points j,i j,i+1 j,i+2... is on a great circle Obtained by projecting bilinear grids to te spere Projection of any vector r to te spere wit r r r image ra: ra = /

16 Bilinear grids Four points r1,r2,r3,r4 may ave any position in space Divide te sides of te romboid equally and connect opposite points Bilinear grid teorem: eac coordinate line intersects eac line of te crossing coordinate line family. Te grid is regular in eac direction.

17 Edges grid (structured) Order 3 or 4 Local coordinate, for example local geograpic irregular, but locally nearly regular grid Non ortogonal Great Circle Grid Stencils Most of te establised concepts of local discretisation in local non-ydrostatic modelling can easily be transferred.

18 Interpolation Grid redundancy is an issue for all metods relying on interpolation Cascade interpolation for regular grids Serendipidity interpolation: te part going into 2d and 3d look like linear. Tird order formula on sides of romboid only. Serendipidity grids replace forecasts of some points by order consistent interpolation

19 Serendipidity grids Grid interpolation: 3rd order (used ere at te boundary) In a square grid grid redundancy is so large (64:15) tat it becomes a problem Some of te redundant grid points can be interpolated from te non redundant ones in an order consistent way, resulting in: efficient interpolation Saving of 27:7 from redundantly forecasted points Easier for posing boundary values and teoretically more satisfying in tis respect Te field is known to 4 rt order error at all points, meaning tat c- grid structures for fast waves can be generated. Spectral elements(principle of minimum overlap for given order leads to serendipidity grids: Some of te basis functions can be sown to contribute in a neglegible way avoiding tese contributions leads to te same grid structures and

20 Semi-implicit integration Semi-implicit integration is important for computer efficient modelling It is a necessary step towards semi implicit. For small areas efficient solvers are available (direct or even SOR) On isocaedral grids different areas may even ave different grid stencils

21 Example: sallow water equations on local coordinates x,y: ) ( y x ST t y STv t v x STu t u + = = = ST: slow tendencies ) ( y x ST t y STv t v x STu t u + = = = Discretisation on staggered C-grid x u u u u u dx x i i = +1

22 Semi implicit integration Semi implicit equs for te sallow water equations in local coordinates x,y: Semi implicit treatment = sl1( u, v, ) + + α *( + n+ 1 n n+ 1 n+ 1 n+ 1 i+ 1 i+ 1 i 1 A omogeneous Helmoltz equation can be acieved assuming: α = const Stability considerations ave to be observed for te splitting involved. Helmoltz solvers love small areas; direct solvers are possible Domain decomposition based on discreet Green functions allow different solution procedures to be used in different areas, even te use of different stencils )

23 Definition of discreet Green function Semi implicit equs: n+ = sl1( u, v, ) + + α *( + 1 n n+ 1 n+ 1 n+ 1 i+ 1 i+ 1 i 1 sl 2( u, v, ) = sl1, for : i = i, = 0 0 : oterwise Green function equation for g: g = α *( g g + g n+ 1 n+ 1 n+ 1 n+ 1 i+ 1 i+ 1 i 1 Te solution for is a linear combination of te g s. Wit cfl = dt / dx we obtain for te Green function 0 ) ) g g = e = e + cfl* x / dx cfl* x / dx or

24 Semi implicit integration: domain decomposition by discreet Green function Solve Helmoltz equation for areas (yellow) define for lines.(red) Use discreet Green function to correct solution in areas Correct solution for areas using Green function compute solution for points (green) Correct solution for areas again Any solution procedure for lines and points will do. If te semiimplicit equations are used as well, a Helmolts equation for te edges grid will arise: Tis Helmolts equation degenerates to one eq for te central point if exp(-cfl*dx/l)<acceptable accuracy.

25 Saving factors of Discretisations Finite Volumes: 1 Baumgardner Order2: 1 Baumgardner Order3: 1 Great circle grids: RK, SI, SL 1 now 3 seem possible Tiled grids: 1.5 Serendipidity grids 3 Unstructured 1/1.3 Conservation 1/2

26 Specifications used for te test model Time discretisation: RK 4 wit no intrinsic or explicit diffusion Fourt order stencil in space Boundary interpolation: tird order on serendipidity grid

27 Results

28 Spectral solution for case 6 and difference to solution on icosaedral grid (5 forecast)

29 Error of x-derivative as function of number of points in x direction

30 Tird order convergence of time derivative of in te sallow water equations

31 2.5 forecasts wit test case 6 (Williamsson) for different resolutions. N96 is used as reference.

32 Conclusions Te approximation order 3+ is possible on isocaedral grids. Energy and mass conservation is a reasonable callenge not yet realised. For ig approximation order it is even a callenge for limited area. For semi implicit time integration te discreet Green function approac is suggested.

33 Dual grid and conservation Use conservation form, compute fluxes 1st possibility: WRF-metod 2nd: Flux correction Issue: order of representation of te conserved quantity